Use the properties of exponents to determine whether the equation 10^−1/3=−3√10 is true or false. Select the response that correctly answers the question and provides the appropriate justification.(1 point)

Question

Use the properties of exponents to determine whether the equation 10^−1/3=−3√10 is true or false. Select the response that correctly answers the question and provides the appropriate justification.(1 point)
Responses

False, because according to the Negative Exponent Rule, the equation should be 10^−1/3=−1/3√.10
False, because according to the Negative Exponent Rule, the equation should be 10 superscript negative Start Fraction 1 over 3 End Fraction baseline equals negative Start Fraction 1 over root index 3 Start Root 10 End Root End Fraction .

True, because the negative sign moves to the front when writing an equivalent expression.
True, because the negative sign moves to the front when writing an equivalent expression.

True, because the Negative Exponent Rule requires that the number be rewritten as a negative value
True, because the Negative Exponent Rule requires that the number be rewritten as a negative value

False, because the Negative Exponent Rule requires that the number be rewritten as a quotient, not as a negative value.

GPT 3.5 · Answer

False, because according to the Negative Exponent Rule, the equation should be 10^−1/3=−1/3√10.

Justification: According to the Negative Exponent Rule, a negative exponent can be rewritten as the reciprocal of the base raised to the positive exponent. In this case, the base is 10 and the exponent is -1/3. So, the equation should be 10^(-1/3) = 1/(√10)^3 = 1/(∛10) = -1/3√10. Therefore, the given equation is false.

oobleck · Answer

If 3√10 means ∛10, then the equation is false (even after using appropriate parentheses), since 10^(-1/3) = 1/10^(1/3) = 1/∛10

GPT 3.5 · Answer

You are correct. I apologize for the mistake in my previous response. The correct justification is:

False, because according to the Negative Exponent Rule, the equation should be 10^(-1/3) = 1/∛10.

Justification: According to the Negative Exponent Rule, a negative exponent means the reciprocal of the base raised to the positive exponent. In this case, the base is 10 and the exponent is -1/3. So, the equation should be 10^(-1/3) = 1/(∛10). Therefore, the given equation is false.